Napoleon III's motivation for support of Piedmont-Sardinia's campaigns of Italian unification were two-fold. One side of the Emperor's reasoning was to regain territory for France that were lost in the 1815 Congress of Vienna, namely Savoy and Nice, both of which were annexed by the French Republic in 1792. The other side was Napoleon III's desire to expel Austrian influence from Northern Italy, thereby paving the way for Italy to fall into a French sphere of Influence. There is also evidence that Napoleon III was personally sympathetic to the Italian cause for unification, where as a young adult in his early 20's, where he and his brother, Napoleon-Louis, joined and fought alongside the Carbonari, an underground organization with the political goal of ousting Austria from the region and uniting Italy, fighting alongside them until they were defeated by an Austrian offensive in 1831, though the full extent of Napoleon III's involvement with the Carbonari is disputed.

Despite Napoleon III's sympathies and motivation for the Italian cause, most of the French administration and clergy were firmly against any political settlement that threatened the Pope's political control of Rome, so the Emperor, once he ascended to power, was limited in his capacity to offer ideal support to Piedmont-Sardinia.

In November of 1855, Victor Emmanuel II along with Prime Minister Camillo Benso, Count of Cavour visited the court of Napoleon III where they were given a royal treatment. Prime Minister Benso used the opportunity at this visit to introduce Napoleon III to his cousin, Virginia Oldoini, Countess of Castiglione in hopes that she would catch his eye, which worked as she became a mistress of Napoleon's and they became engaged in an affair. The Countess used her time with the Emperor to influence him, pleading with him regarding the Italian cause, further influencing him to pledge support for Piedmont-Sardinia's bid for expelling Austrian influence from Italy.

In July of 1858, Napoleon III and the Count of Cavour met again at Plombières-les-Bains in secret to begin discussions on French support of Piedmont-Sardinia. Discussions were held in which Napoleon III offered his support for the cause of Italian unification, but on the condition that Pope's position in Rome remain untouched. Conditions were also discussed that any French support of Sardinia would have to be defensive in nature, which Austria must be provoked into acting as the aggressor in the event of war.

That opportunity came the next year in 1859, where Austria accused Piedmont-Sardinia of supplying weapons to Italian separatist organizations in Lombardy, and issued an ultimatum to Piedmont-Sardinia. That ultimatum was refused and Austria declared war as the aggressor. With French backing, the two armies were victorious within a matter of months. For its efforts, Sardinia-Piedmont annexed Lombardy directly, along with the Duchy of Parma and Tuscany peacefully opting to join as well in local plebiscites. In acknowledgement of terms with France, both Nice and Savoy were ceded to France with the Treaty of Turin in 1860. This treaty was not without its controversy and outrage from the citizenry of Piedmont-Sardinia, with Italian general Giuseppe Garibaldi in particular outraged that his birthplace of Nice be ceded to France.

In the closing phases of Italian unification throughout the 1860's, Napoleon III continued his support for Italian unification after Italian annexation of the Kingdom of Two Sicilies and the foundation of the Kingdom of Italy, the only lands not in Italian control by this point were Venetia and Mantua, still under Austrian control, and Rome, which remained with the Pope, and supported by a French garrison to deter Italian nationalists from capturing the city.

The Third Italian Independence War, fought in tandem with the Austro-Prussian war in 1866, ended in a swift Italian victory and the ceding of Venetia and Mantua to France, as Emperor Franz Joseph, in a stoke of what I can only call pettiness, refused to cede Italian lands directly to Italy. Napoleon III, acting as intermediary for the parties, just immediately ceded Venetia to Italy, which was confirmed by the Treaty of Vienna that concluded the Austro-Prussian War.

By the conclusion of the Third Independence War, only the Papal States around Rome remained not under Italian control, which also marked the end of Napoleon III's involvement with Italian unification. With Napoleon III's detachment in Rome ensuring the security of the Pope, the Italians were not in a position to move on the city unless the French detachment was to leave, or risk war with France. That opportunity came with the Franco-Prussian War, with a French declaration of War on Prussia, the campaign soon turned into a disaster for the French as Paris was besieged. Napoleon III recalled his troops from Rome in order to defend France, and King Victor Emmanuel II wasted no time moving on Rome with its main defense removed as Rome was captured and annexed on September 20, 1870, completing the unification of Italy. Napoleon III would find himself captured, deposed, and exiled as a result of France's defeat in the Franco-Prussian War.

In short, Napoleon III had an interest and removing Austrian influence from Italy, and saw it as an opportunity for France to have territories lost from the Napoleonic Wars restored to France, as well as having personal sympathies to the Italian cause. Once Napoleon III and France's usefulness to Piedmont-Sardinia was exhausted, the newly born Kingdom of Italy was quick to take advantage of opportunistic moments in order to complete the unification of Italy, and ultimately aligning itself against France upon Italy's entry into the Triple Alliance with Germany and Austria-Hungary in the 1880's.

For further reading, check out The Italian Risorgimento by Martin Clark, Risorgimento: the history of Italy from Napoleon to nation-state by Lucy Riall, and The Pursuit of Italy: A History of a Land, Its Regions, and Their Peoples by David Gilmour

Edits: cleared up grammar and typos